Divorce was allowed in Babylonian society. The most common reasons being infertility on the wife’s part (the Husband was never to blame!) or infidelity, which was always more severely dealt with if the wife had been at fault,

Didn't Hammurabi's laws permit the man to be sexually active outside of marriage? (but not the woman)

I guess that would depend on what kind of sex he had.  If he had multiple wives yes.  If he had a concubine yes.  If he had sex with vestal virgin type girls yes.  If he committed fornication/adultery probably not....but I doubt too many men got "caught" doing that anyway.

ny24v0